MEN OF MATHEMATICS To see that these are reasonable, consider for example the second, ab = a. This states that if a is included in &, then every- thing that is hi both a and b is the whole of a. From the stated postulates the following theorems on inehi- sion (with thousands of more complicated ones, if desired) can be proved. The specimens selected all agree with our intuitive conception of what ^inclusion' means. (1) a < a. (2) lfa<b and b < c, then a < c. (3) Ifa<b and b < a, then a = b. (4) Z < a (where Z is the element in II a - it is proved to be the only element satisfying II a). (5) a < U (where U is the element in II b - likewise unique). (6) a < a + b-, and ifa<y and b <ys then a + b < y. (7) ab < a; andifx < aandx < b, thenx < ab. (8) Ifx < a and x < a', then x = Z; andifa < y and a' < y, then y = U. (9) If a < br is false, then there is at least one element x, distinct from Z> such that x < a and x < b. It may be of interest to observe that4 <' in arithmetic and analysis is the symbol for less than'. Note that if a, &, e,... are real numbers, and 2 denotes zero, then (2) is satisfied for this interpretation of '<', and similarly for (4), provided a is posi- tive; but that (1) is not satisfied, nor is the second part of (6) - as we see from 5 < 10,7 < 10, but 5 4- 7 < 10 is false. The tremendous power and fluent ease of the method can be readily appreciated by seeing what it does in any work on symbolic logic. But, as already emphasized, the importance of this 'symbolic reasoning* is hi its applicability to subtle ques- tions regarding the foundations of all mathematics which, were it not for this precise method of fixing meanings of 'words' or other 'symbols* once for all, would probably be unapproachable by ordinary mortals. Like nearly all novelties, symbolic logic was neglected for many years after its invention. As late as 1910 we find eminent mathematicians scorning it as a 'philosophical' curiosity with- out mathematical significance. The work of Whitehead and Russell in Principia Mathematica (1910-13) was the first to 492